Zubov-Net: Neural Methods for ROA Estimation

• Zubov-Net is a family of neural network methods that use Zubov theory to characterize regions of attraction (ROA) and safety domains in nonlinear dynamical systems.
• These approaches approximate Zubov-type PDEs, Hamilton–Jacobi–Bellman formulations, or Koopman fixed-point relations to enable controller synthesis, safety certification, and robustness regularization.
• Implementations range from two-stage neural controller designs and PINN-based safety verifications to operator learning techniques, yielding improved computational efficiency and accuracy.

Searching arXiv for the specified papers and closely related work on Zubov-Net. Zubov-Net is a label used in recent literature for a family of neural-network methods derived from Zubov theory for nonlinear dynamical systems. Across these works, the central object is a learned scalar function—variously denoted $W$, $\tilde W_\theta$, $V_\phi$, or $W_\theta$—whose sublevel geometry encodes a region of attraction (ROA), a safe domain of attraction, or a safe-stabilizable domain. The common technical motif is to approximate either a Zubov partial differential equation, a Zubov–Hamilton–Jacobi–Bellman equation, or a Zubov–Koopman fixed-point relation, and then use the learned function for controller synthesis, safety certification, robustness regularization, or ROA estimation (Li et al., 2 Jun 2025, Meng et al., 12 Nov 2025, Meng et al., 1 Apr 2026, Luo et al., 26 Sep 2025, Meng et al., 2023).

1. Terminological scope and problem setting

In the cited literature, “Zubov-Net” does not denote a single canonical architecture. Instead, it names several related constructions that all use Zubov-style characterizations of attraction domains. One line of work presents a two-stage framework that jointly synthesizes a neural state-feedback controller $u_\theta(x)$ and a Lyapunov-like function $V_\phi(x)$ for continuous-time systems, with direct ROA estimation and formal verification via an extended $\alpha,\!\beta$-CROWN verifier (Li et al., 2 Jun 2025). A second line treats “Zubov-Net” as a physics-informed neural network (PINN) for solving a Dirichlet-form Zubov PDE on a safe set boundary, so that the resulting function acts as a Lyapunov-barrier certificate (Meng et al., 12 Nov 2025). A third line extends the underlying PDE viewpoint to control-affine safe stabilization through a Zubov–HJB formulation whose viscosity solution yields a maximal control Lyapunov-barrier function (CLBF) (Meng et al., 1 Apr 2026). A fourth line uses the name for an adaptive stable learning framework for Neural ODE classifiers, where Zubov’s equation is reformulated as a consistency condition between prescribed regions of attraction (PRoAs) and true ROAs (Luo et al., 26 Sep 2025). A fifth, earlier line uses the term for a neural approximation of a Zubov–Koopman operator, learned from data and iterated to recover a Zubov solution for an unknown system (Meng et al., 2023).

The unifying task is the constructive approximation of a function whose level sets separate asymptotically stable behavior from instability or unsafety. In the classical autonomous setting, the exact ROA is identified with $\{x:W(x)<1\}$; in the safety-constrained setting, the same structure is applied on a safe set $D$ with Dirichlet data $W=1$ on $\tilde W_\theta$0; in the control-affine setting, a maximization over admissible controls enters the PDE; and in the Neural ODE classification setting, each class equilibrium $\tilde W_\theta$1 is equipped with its own $\tilde W_\theta$2, whose sublevel set $\tilde W_\theta$3 is interpreted as a PRoA (Li et al., 2 Jun 2025, Meng et al., 12 Nov 2025, Meng et al., 1 Apr 2026, Luo et al., 26 Sep 2025).

2. Zubov-theoretic foundations

For continuous-time closed-loop dynamics

$\tilde W_\theta$4

Zubov’s theorem characterizes the true ROA $\tilde W_\theta$5 by a continuously differentiable function $\tilde W_\theta$6 such that $\tilde W_\theta$7, $\tilde W_\theta$8 for $\tilde W_\theta$9, $V_\phi$0 as $V_\phi$1 or $V_\phi$2, and

$V_\phi$3

A practical choice reported from Liu et al. is

$V_\phi$4

which yields

$V_\phi$5

Imposing $V_\phi$6 on $V_\phi$7 pins down the maximal ROA exactly as $V_\phi$8 (Li et al., 2 Jun 2025).

An equivalent autonomous formulation writes

$V_\phi$9

with $W_\theta$0 and $W_\theta$1 on $W_\theta$2, where $W_\theta$3 is the domain of attraction. In that representation, $W_\theta$4 satisfies

$W_\theta$5

and the ROA is exactly

$W_\theta$6

This is the form used in the Zubov–Koopman lifting approach (Meng et al., 2023).

For safety-constrained autonomous systems, the literature introduces a Dirichlet-form Zubov PDE on the safe domain $W_\theta$7: $W_\theta$8 with boundary conditions

$W_\theta$9

On a larger compact ROI $u_\theta(x)$0, a positively rescaled modified problem is written as

$u_\theta(x)$1

where $u_\theta(x)$2 (Meng et al., 12 Nov 2025).

For control-affine systems

$u_\theta(x)$3

the Zubov construction becomes an HJB-type PDE. If $u_\theta(x)$4 is the value function defined by an infinite-horizon running cost $u_\theta(x)$5, then $u_\theta(x)$6 is the unique viscosity solution of

$u_\theta(x)$7

on the maximal safe-stabilizable domain $u_\theta(x)$8, with $u_\theta(x)$9 and $V_\phi(x)$0 as $V_\phi(x)$1 or $V_\phi(x)$2. Under the Zubov transformation $V_\phi(x)$3, one obtains

$V_\phi(x)$4

with $V_\phi(x)$5 and $V_\phi(x)$6 on $V_\phi(x)$7 (Meng et al., 1 Apr 2026).

These formulations all preserve the same geometric principle: if the exact solution is obtained, then $V_\phi(x)$8 is the maximal attractor-compatible domain. This suggests that the principal source of variation among Zubov-Net methods lies not in the underlying theorem, but in how the equation is approximated, sampled, regularized, and verified.

3. Two-stage neural controller synthesis and formal certification

The controller-synthesis variant parameterizes both a controller and a Zubov function by multilayer perceptrons. The controller is

$V_\phi(x)$9

where $\alpha,\!\beta$0 is the known symmetric input limit and $\alpha,\!\beta$1 is the equilibrium input. Each output coordinate is individually clamped to $\alpha,\!\beta$2. The Zubov network is

$\alpha,\!\beta$3

with $\alpha,\!\beta$4, ensuring $\alpha,\!\beta$5. In practice, both $\alpha,\!\beta$6 and $\alpha,\!\beta$7 are standard fully connected layers with $\alpha,\!\beta$8–$\alpha,\!\beta$9 hidden layers, width $\{x:W(x)<1\}$0–$\{x:W(x)<1\}$1, tanh or ReLU hidden activations, and no batch normalization (Li et al., 2 Jun 2025).

Training proceeds in two stages. Stage 1 performs ROA estimation through a Zubov-guided curriculum. Every $\{x:W(x)<1\}$2 steps, an UpdateDomain procedure expands the current domain $\{x:W(x)<1\}$3. Two batches of $\{x:W(x)<1\}$4 points are sampled by “Zubov-guided PGD”: interior points minimize $\{x:W(x)<1\}$5, and exterior points minimize $\{x:W(x)<1\}$6. Boundary points are sampled on $\{x:W(x)<1\}$7. The composite loss includes $\{x:W(x)<1\}$8, a Zubov PDE residual $\{x:W(x)<1\}$9, a trajectory-consistency term $D$0, a controller term $D$1, and a boundary term $D$2, with weights learned via uncertainty weighting. The UpdateDomain procedure samples $D$3 points in $D$4 by PGD, simulates them for time $D$5 at step $D$6, records dimensionwise minima and maxima of convergent trajectories, forms a new box $D$7, and expands it by factor $D$8 (Li et al., 2 Jun 2025).

Stage 2 is a CEGIS refinement stage. It maintains a small buffer of counterexamples, attacks points in $D$9 via PGD to maximize

$W=1$0

and then descends

$W=1$1

plus a regularizer

$W=1$2

Training stops when no new counterexamples appear. The stated intuition is that Stage 1 rapidly learns a roughly correct ROA by combining a physics-informed Zubov PDE loss with interior/exterior sampling and domain expansion, and Stage 2 removes local Lyapunov violations inside the estimated level set (Li et al., 2 Jun 2025).

Formal certification is based on an extension of $W=1$3-CROWN to continuous-time Jacobian products. To certify forward invariance of $W=1$4, the method uses a theorem requiring, for some $W=1$5, negativity of $W=1$6 on $W=1$7 and inward-pointing vector field behavior on $W=1$8. The verifier adds tight linear relaxations for $W=1$9 and $\tilde W_\theta$00 over input ranges $\tilde W_\theta$01, supports these in the bound-propagation engine, and replaces repeated bisection on $\tilde W_\theta$02 with an adaptive branch-and-bound scheme that updates $\tilde W_\theta$03 when a small counterexample is found, without re-verifying already certified subdomains (Li et al., 2 Jun 2025).

Reported quantitative results are explicit. ROA volume improvements over baselines are up to $\tilde W_\theta$04 on $\tilde W_\theta$05D systems, $\tilde W_\theta$06 on Cartpole ($\tilde W_\theta$07D), $\tilde W_\theta$08 on PVTOL ($\tilde W_\theta$09D), and $\tilde W_\theta$10 on a $\tilde W_\theta$11D quadrotor, with an overall estimate of up to $\tilde W_\theta$12 via Monte Carlo in $\tilde W_\theta$13D. On $\tilde W_\theta$14D benchmarks, dReal takes $\tilde W_\theta$15–$\tilde W_\theta$16 s, whereas the extended CROWN verifier finishes each in $\tilde W_\theta$17–$\tilde W_\theta$18 s, corresponding to a $\tilde W_\theta$19–$\tilde W_\theta$20 speedup. The same source states that dReal often fails on Cartpole, while CROWN verifies it in approximately $\tilde W_\theta$21 s. It also notes that beyond $\tilde W_\theta$22D, Jacobian bound tightness remains a challenge; in $\tilde W_\theta$23D, forward invariance is verified only for a thin band, and empirical PGD or Monte Carlo is used for higher dimensions (Li et al., 2 Jun 2025).

4. Dirichlet Zubov-Net for safety and maximal CLBFs

The safety-oriented PINN formulation learns a function $\tilde W_\theta$24 on a compact ROI $\tilde W_\theta$25 using a fully connected feed-forward network with two hidden layers of width $\tilde W_\theta$26 and smooth activations such as tanh. Training uses $\tilde W_\theta$27 collocation points in $\tilde W_\theta$28, $\tilde W_\theta$29 boundary points on $\tilde W_\theta$30, and the origin. The losses are

$\tilde W_\theta$31

$\tilde W_\theta$32

and the total loss is

$\tilde W_\theta$33

The reported hyperparameters are $\tilde W_\theta$34, $\tilde W_\theta$35, $\tilde W_\theta$36, with Latin-Hypercube sampling for collocation points, uniform sampling on $\tilde W_\theta$37, $\tilde W_\theta$38, $\tilde W_\theta$39, and Adam for $\tilde W_\theta$40 epochs ($\tilde W_\theta$41k gradient steps) at learning rate $\tilde W_\theta$42 (Meng et al., 12 Nov 2025).

Verification in this framework targets both forward invariance and obstacle avoidance. For a sublevel set

$\tilde W_\theta$43

the goal is to prove

$\tilde W_\theta$44

LyZNet calls dReal or Z3 to prove the first-order formulas

$\tilde W_\theta$45

and

$\tilde W_\theta$46

for some small margin $\tilde W_\theta$47. Under the corresponding lemma, if $\tilde W_\theta$48 on $\tilde W_\theta$49 and $\tilde W_\theta$50, then $\tilde W_\theta$51 is forward-invariant and contains no unsafe points (Meng et al., 12 Nov 2025).

This safe-domain formulation is closely related to the later control-affine Zubov–HJB theory. There, the maximal safe-stabilizable domain is

$\tilde W_\theta$52

under a compatibility assumption near the safe boundary. The transformed solution $\tilde W_\theta$53 is shown to be a continuous viscosity solution and a nonsmooth CLBF, with $\tilde W_\theta$54. Feedback is synthesized pointwise by

$\tilde W_\theta$55

or equivalently in Zubov form using $\tilde W_\theta$56. The associated “Zubov-Net” implementation recipe minimizes interior PDE residual, Dirichlet boundary losses, optional inequality penalties, and a gradient regularizer, using a fully connected MLP with tanh or softplus and either a sigmoid output or a clamped parameterization to keep $\tilde W_\theta$57 (Meng et al., 1 Apr 2026).

The empirical benchmark reported for the Dirichlet safe-domain PINN is the reversed Van der Pol system with two circular obstacles, where the learned and verified level set recovered roughly $\tilde W_\theta$58 of the true area, while a quadratic CLF approach captured $\tilde W_\theta$59. Training times are reported as $\tilde W_\theta$60–$\tilde W_\theta$61 minutes and formal verification as $\tilde W_\theta$62–$\tilde W_\theta$63 minutes per example. The same source identifies two limitations: the need to choose a proper indicator $\tilde W_\theta$64 such that $\tilde W_\theta$65 at $\tilde W_\theta$66 and $\tilde W_\theta$67 diverges, and the fact that formal verification can be conservative and does not scale beyond $\tilde W_\theta$68–$\tilde W_\theta$69 (Meng et al., 12 Nov 2025).

5. Zubov-Net for Neural ODE robustness and prescribed attraction geometry

A distinct use of the name appears in Neural ODE classification, where Zubov-Net is an “adaptive stable learning framework” that seeks to reconcile robustness and accuracy by actively controlling basin geometry. The system is

$\tilde W_\theta$70

with equilibrium set $\tilde W_\theta$71, one equilibrium per class. The true ROA of equilibrium $\tilde W_\theta$72 is

$\tilde W_\theta$73

and the PRoA is defined by a learnable Lyapunov function $\tilde W_\theta$74 through

$\tilde W_\theta$75

The stated interpretation is that $\tilde W_\theta$76 is the current guess of $\tilde W_\theta$77 (Luo et al., 26 Sep 2025).

The central reformulation is a consistency loss derived from Zubov’s equation. If $\tilde W_\theta$78, $\tilde W_\theta$79 on $\tilde W_\theta$80, and

$\tilde W_\theta$81

for all $\tilde W_\theta$82, with $\tilde W_\theta$83 for $\tilde W_\theta$84, then $\tilde W_\theta$85. The paper turns this into

$\tilde W_\theta$86

and globally enforces

$\tilde W_\theta$87

It states that $\tilde W_\theta$88 if and only if each PRoA exactly satisfies Zubov’s PDE and hence aligns with the true $\tilde W_\theta$89 (Luo et al., 26 Sep 2025).

Two additional losses control class assignment and basin separation. The classification term uses the Lyapunov-value vector

$\tilde W_\theta$90

and the normalized inverse-value map

$\tilde W_\theta$91

with

$\tilde W_\theta$92

The separation term samples boundary points $\tilde W_\theta$93 and penalizes

$\tilde W_\theta$94

where $\tilde W_\theta$95 and $\tilde W_\theta$96. The total objective combines classification, a term $\tilde W_\theta$97, consistency, and separation (Luo et al., 26 Sep 2025).

A notable architectural contribution is the input-attention convex neural network for the pre-Lyapunov potential

$\tilde W_\theta$98

The $\tilde W_\theta$99-layer IACNN uses softmax attention over $V_\phi$00 and previous features $V_\phi$01, with elementwise nonnegative $V_\phi$02 and convex non-decreasing $V_\phi$03, so that $V_\phi$04 is convex in $V_\phi$05. The paper states that the softmax attention mechanism focuses on equilibrium-relevant features and also serves as weight normalization to maintain training stability in deep architectures (Luo et al., 26 Sep 2025).

The parallel boundary sampling algorithm updates radial lengths $V_\phi$06 along directions $V_\phi$07 by sign feedback until the target level $V_\phi$08 is reached. All updates are performed in parallel, and the argument relies on radial monotonicity induced by strong convexity. The stated theoretical guarantees are: $V_\phi$09 if and only if $V_\phi$10; $V_\phi$11 implies $V_\phi$12 for $V_\phi$13; and if $V_\phi$14 and $V_\phi$15, then the entire trajectory $V_\phi$16 for $V_\phi$17 stays in $V_\phi$18 (Luo et al., 26 Sep 2025).

The empirical results are reported on SVHN, CIFAR-10, and CIFAR-100. For noise robustness, average over eight corruptions plus clean is $V_\phi$19 on SVHN versus $V_\phi$20 for the next best SODEF, $V_\phi$21 on CIFAR-10 versus $V_\phi$22, and $V_\phi$23 on CIFAR-100 versus $V_\phi$24. For white-box adversarial robustness, averaged over FGSM, PGD, BIM, APGD, and Jitter at $V_\phi$25, the results are $V_\phi$26 on SVHN versus $V_\phi$27 for FxTS-Net, $V_\phi$28 on CIFAR-10 versus $V_\phi$29, and $V_\phi$30 on CIFAR-100 versus $V_\phi$31. Clean accuracy remains on par or better than baselines, with CIFAR-10 reported as $V_\phi$32 versus $V_\phi$33 for a standard Neural ODE. The ablation claims are explicit: removing $V_\phi$34 sharply degrades adversarial robustness, removing $V_\phi$35 degrades noise robustness, and the full tripartite loss gives the best overall trade-off (Luo et al., 26 Sep 2025).

6. Zubov–Koopman lifting and data-driven ROA learning

The Zubov–Koopman line addresses unknown autonomous systems from data. For

$V_\phi$36

with asymptotically stable equilibrium at $V_\phi$37, the one-step Zubov–Koopman operator is defined for any bounded continuous $V_\phi$38 by

$V_\phi$39

where $V_\phi$40 is the flow and $V_\phi$41 is a positive-definite weight. The family $V_\phi$42 is a strongly continuous linear semigroup on $V_\phi$43, with infinitesimal generator

$V_\phi$44

The Zubov solution $V_\phi$45, extended by $V_\phi$46 outside the ROA, is the unique bounded fixed point of each one-step map: $V_\phi$47 This converts ROA estimation into operator learning (Meng et al., 2023).

The neural parameterization chooses feature functions

$V_\phi$48

and approximates

$V_\phi$49

with $V_\phi$50. A decoder row vector $V_\phi$51 then yields

$V_\phi$52

Under power iteration, the full ANN is

$V_\phi$53

which the source calls the “Zubov-Net.” Typical choices include sinusoidal, Fourier, monomial, or small-MLP features; $V_\phi$54 may be unconstrained or regularized; and $V_\phi$55 may use softplus to enforce $V_\phi$56 (Meng et al., 2023).

Training combines an EDMD-style operator-fit term and a PDE residual. Given pairs $V_\phi$57 and weights $V_\phi$58, the losses are

$V_\phi$59

and

$V_\phi$60

with total loss

$V_\phi$61

The recovery step uses

$V_\phi$62

and under a spectral gap assumption on $V_\phi$63, $V_\phi$64 as $V_\phi$65. The final Lyapunov candidate is $V_\phi$66 (Meng et al., 2023).

The paper states several convergence ingredients: the semigroup property $V_\phi$67; the improper-integral formula

$V_\phi$68

yielding a bounded continuous function; uniqueness of the bounded viscosity solution of the Zubov PDE by a comparison principle; finite-rank approximation of $V_\phi$69 by projection onto leading eigenfunctions; EDMD strong convergence with sufficiently rich dictionary and enough data; and convergence of $V_\phi$70 to the fixed-point projector when the learned $V_\phi$71 has dominant eigenvalue $V_\phi$72 and spectral gap (Meng et al., 2023).

The numerical examples include reversed Van der Pol, a polynomial stiff system, a $V_\phi$73D nonlinear system, and a two-machine power system with a sine nonlinearity. The common setup uses a region $V_\phi$74 containing the ROA, $V_\phi$75, $V_\phi$76 or a minor variant, a small MLP dictionary with two hidden layers of $V_\phi$77 units and ReLU or smooth activations, batch size $V_\phi$78–$V_\phi$79, and $V_\phi$80–$V_\phi$81 epochs of joint EDMD plus PDE loss followed by $V_\phi$82–$V_\phi$83 power iterations. In the reversed Van der Pol example, the Hausdorff error is approximately $V_\phi$84. The report states that all examples converge in under $V_\phi$85 power iterations and are finally checked by verifying $V_\phi$86 on a dense grid in $V_\phi$87 (Meng et al., 2023).

7. Limitations, misconceptions, and research directions

A common misconception is that Zubov-Net denotes a single fixed neural architecture. The literature instead supports a broader interpretation: the term refers to several neural approximators of Zubov-type objects—PDE solutions, viscosity solutions, Lyapunov-barrier functions, or operator fixed points—adapted to different tasks such as controller synthesis, safety certification, Neural ODE robustness, or operator learning (Li et al., 2 Jun 2025, Meng et al., 12 Nov 2025, Luo et al., 26 Sep 2025, Meng et al., 2023).

Another misconception is that all Zubov-Net methods are verification-complete. The sources distinguish sharply between approximate learning and formal certification. The controller-synthesis framework extends $V_\phi$88-CROWN and reports substantial speedups over dReal, but also states that Jacobian bound tightness remains a challenge beyond $V_\phi$89D and that empirical PGD or Monte Carlo is used in higher dimensions (Li et al., 2 Jun 2025). The safe-domain PINN framework uses SMT solvers through LyZNet, yet explicitly notes that formal verification is conservative and does not scale beyond $V_\phi$90–$V_\phi$91 (Meng et al., 12 Nov 2025). The Neural ODE robustness framework proves consistency, non-overlap, and finite-time containment under the exact-loss condition $V_\phi$92, but its empirical claims concern robustness metrics rather than formal reachability certificates (Luo et al., 26 Sep 2025). The Koopman-lifting approach provides strong convergence statements under regularity, richness, and spectral assumptions, but its final verification step is described as dense-grid checking rather than theorem-proving (Meng et al., 2023).

The main technical bottlenecks are also consistent across the literature. High-dimensional verification is limited by bound tightness for Jacobian terms in continuous-time systems (Li et al., 2 Jun 2025). Boundary-layer resolution may require many collocation points in very high dimensions (Meng et al., 12 Nov 2025). Safe-domain formulations require a proper indicator $V_\phi$93 whose blow-up behavior is suitable near the boundary (Meng et al., 12 Nov 2025). Control-affine formulations rely on compatibility assumptions near the safe boundary and on viscosity-solution machinery, which suggests that numerical realization is naturally more delicate than the autonomous case (Meng et al., 1 Apr 2026).

Taken together, these works position Zubov-Net not as a single algorithm, but as a research program centered on constructive approximations of maximal attraction-compatible functions. The consistent structural claim is that, when the relevant Zubov-type equation is solved accurately enough and the resulting certificate is formally or empirically validated, the sublevel set $V_\phi$94 provides a non-conservative under-approximation—or in the ideal exact case, the exact characterization—of the underlying ROA, safe domain of attraction, or safe-stabilizable region (Li et al., 2 Jun 2025, Meng et al., 12 Nov 2025, Meng et al., 1 Apr 2026).